Multiple choice

If $n$ is a positive integer and $n\in[5,100]$, then the number of integral roots of the equation $x^{2}+2x-n=0$ are

  1. $4$
  2. $6$
  3. $8$
  4. $10$
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C Correct answer
Explanation

The equation x^2 + 2x - n = 0 has integral roots if the discriminant D = 4 + 4n is a perfect square, say k^2. This implies 4(1 + n) = k^2, so 1 + n must be a perfect square, m^2. For n in [5, 100], 1 + n is in [6, 101], so m^2 can be 9, 16, 25, 36, 49, 64, 81, 100, which gives 8 possible values for n.

AI explanation

Using the quadratic formula, the roots of x^2 + 2x - n = 0 are x = -1 + sqrt(1 + n) and x = -1 - sqrt(1 + n). For both roots to be integers, 1 + n must be a perfect square, meaning n + 1 = k^2 and n = k^2 - 1 for some integer k. Checking the given range for n from 5 to 100 reveals the perfect squares k^2 from 9 to 101, which correspond to k values of 3, 4, 5, 6, 7, 8, 9, and 10. This provides 8 distinct values for n, resulting in 8 integral roots.

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